US 6655803 B1 Abstract A method for designing a surface of an optical element includes the steps of prescribing initial wavefronts, selecting an initial parameterized representation of the surface, choosing a cost function in the parameters, and optimizing the cost function with respect to the parameters. The step of optimizing includes the steps of calculating a refracted wavefront for each of the initial wavefronts and analytically computing derivatives of the cost function. The method may also include the steps of precomputing eikonal functions between points in the vicinity of the initial wavefronts. Then the refracted wavefronts are calculated from the precomputed eikonal functions. A method for designing at least one surface of a multifocal optical element includes the step of concurrently considering in a design of the at least one surface any combination of an astigmatism distribution, a progressive power distribution and a prism distribution. When designing at least one surface of an optical element, a representation for each particular surface is created with a set of discrete points. Then for each discrete point in the set, nearest neighbor points from the set are selected, the particular surface is approximated in a vicinity of the discrete point by a polynomial of a predetermined order in two variables of the particular surface, and the coefficients of the polynomial are determined according to the selected nearest neighbor points without requiring continuity between polynomials for neighboring discrete points.
Claims(33) 1. A method for designing at least one surface of an ophthalmic lens, the method comprising the steps of:
representing said at least one surface with parameters;
choosing a function in said parameters, said function including a term involving the difference between the astigmatism induced by said ophthalmic lens and a predetermined astigmatism distribution, said difference being raised to an exponent having a value other than 2; and
optimizing said function with respect to said parameters.
2. The method of
3. An ophthalmic lens having two or more surfaces, wherein at least one of said surfaces is designed by the method of
4. A method for designing at least one surface of an ophthalmic lens, the method comprising the steps of:
receiving a prescription of a specific wearer for said ophthalmic lens;
representing said at least one surface with parameters;
choosing a function in said parameters, said function including a term involving the difference between the prism induced by said ophthalmic lens and a predetermined prism distribution related to said prescription; and
optimizing said function with respect to said parameters.
5. The method of
6. An ophthalmic lens having two or more surfaces, wherein at least one of said surfaces is designed by the method of
7. A method for designing at least one surface of an ophthalmic lens, the method comprising the steps of:
representing said at least one surface with parameters;
choosing a function in said parameters, said function including a term involving the difference between the gradient of a characteristic and the gradient of a predetermined characteristic distribution, said characteristic selected from a group including: optical power, optical astigmatism, and optical prism; and
optimizing said function with respect to said parameters.
8. The method of
9. An ophthalmic lens having two or more surfaces, wherein at least one of said surfaces is designed by the method of
10. A method for designing at least one surface of an ophthalmic lens, the method comprising the steps of:
representing said at least one surface with parameters;
choosing a function in said parameters, said function including the square of the difference between the power induced by said ophthalmic lens and a predetermined power distribution and other terms related to the power induced by said ophthalmic lens; and
optimizing said function with respect to said parameters.
11. The method of
12. The method of
_{v}(l,m)|^{β} for at least one value of β other than 2, where P(l,m) denotes the power induced by the optical element, P_{v}(l,m) denotes the predetermined power distribution, and (l,m) parameterize the surfaces of said ophthalmic lens.13. An ophthalmic lens having two or more surfaces, wherein at least one of said surfaces is designed by the method of
14. A method for designing at least one surface of an ophthalmic lens, the method comprising the steps of:
representing said at least one surface by parameters;
choosing an astigmatism distribution and an astigmatism direction distribution;
choosing a function in said parameters, said function including a term involving said astigmatism distribution, said astigmatism direction distribution and the astigmatism induced by said ophthalmic lens; and
optimizing said function with respect to said parameters.
15. The method of
16. The method of
17. The method of
where (l,m) index said parameters for said at least one surface, C
_{v,1 }is said astigmatism distribution, C_{v,2 }is a second astigmatism distribution, α_{11}, α_{12 }and α_{22 }are the coefficients in a quadratic expansion of a wavefront refracted by said lens in a coordinate system chosen for each (l,m) in accordance with said astigmatism direction distribution, w_{1 }and w_{2 }are weight distributions and β is an exponent.18. An ophthalmic lens having two or more surfaces, wherein at least one of said surfaces is designed by the method of
19. The method of
20. The method of
21. The method of
22. An ophthalmic lens having two or more surfaces, wherein at least one of said surfaces is designed by the method of
23. A method for designing at least one surface of an ophthalmic lens, the method comprising the steps of:
representing said at least one surface by a method of unconstrained patches to obtain a representation having parameters, said method of unconstrained patches including the steps of:
selecting a set of discrete points {p
_{i}=(X_{i},y_{i},z_{i})}; approximating said surface in the vicinity of each discrete point p
_{i }by a local polynomial of a predetermined order in the variables x and y; and choosing coefficients of said local polynomial that best fit the discrete values of the nearest neighbors to p
_{i }in the set; choosing a function in said parameters, said function including a weighted combination of terms involving the difference between the astigmatism induced by said ophthalmic lens and a predetermined astigmatism distribution and terms involving the difference between the power induced by said ophthalmic lens and a predetermined power distribution; and
optimizing said function with respect to said parameters.
24. An ophthalmic lens having two or more surfaces, wherein at least one of said surfaces is designed by the method of
25. A method for designing at least one surface of an ophthalmic lens, the method comprising the steps of:
representing said at least one surface by parameters;
choosing a function in said parameters, said function including a weighted combination of terms involving the difference between the astigmatism induced by said ophthalmic lens and a predetermined astigmatism distribution and terms involving the difference between the power induced by said ophthalmic lens and a predetermined power distribution; and
optimizing said function with respect to said parameters, wherein said step of optimizing comprises the steps of:
selecting initial values of said parameters to attain an initial lens, wherein said initial lens includes any fixed surfaces of said ophthalmic lens and said at least one surface with said initial values for said parameters;
computing eikonal functions for said initial lens; and
computing from said eikonal functions optical characteristics of said initial lens and optical characteristics of lenses including said fixed surfaces and said at least one surface with partially optimized values for said parameters.
26. An ophthalmic lens having two or more surfaces, wherein at least one of said surfaces is designed by the method of
27. A method for designing at least one surface of an ophthalmic lens having two or more surfaces, the method comprising the steps of:
representing said at least one surface by parameters;
choosing a function in said parameters, said function including a term involving the difference between the astigmatism induced by said ophthalmic lens and a predetermined astigmatism distribution and a term involving the difference between the power induced by said ophthalmic lens and a predetermined power distribution; and
optimizing said function with respect to said parameters subject to one or more constraints selected from a group including: constraining said surfaces from becoming too close to one another, and constraining said at least one surface or its derivatives or both to prescribed values at a predetermined set of points.
28. An ophthalmic lens having two or more surfaces, wherein at least one of said surfaces is designed by the method of
29. A method for simultaneously designing a front surface and a back surface of an ophthalmic lens, the method comprising the steps of:
selecting chief rays, each of said chief rays associated with a viewing angle of a point object at a predefined distance from a central point in an eye;
representing said back surface by a first set of distances, each of said first set of distances being the distance of said central point from an intersection point of said back surface with a respective one of said chief rays;
representing said front surface by a second set of distances, each of said second set of distances being the distance of an intersection point of said front surface with a respective one of said chief rays from the intersection point of said back surface with said respective one of said chief rays;
expressing said first set of distances and said second set of distances as parameterized functions of said viewing angle;
choosing a cost function in parameters of said parameterized functions, said cost function including a weighted combination of one or more terms involving the astigmatism induced by said ophthalmic lens and a predetermined astigmatism distribution and one or more terms involving the power induced by said ophthalmic lens and a predetermined power distribution; and
optimizing said cost function with respect to said parameters.
30. The method of
31. The method of
32. An ophthalmic lens having a front surface and a back surface both of which are designed simultaneously by the method of
33. The ophthalmic lens of
Description The present invention relates to the design of optical elements, in general, and to a method for designing optical elements using wavefront calculations, in particular. Most of the designs for progressive (multifocal) ophthalmic lenses are based mainly on the geometrical shape of one or more of the lens surfaces, as described, for example in U.S. Pat. No. 3,687,528 to Maitenaz. The geometrical properties, however, are only indirectly related to the lens' actual optical performance. For example, the curvature of any of the lens surfaces is only approximately related to the lens optical power. In some cases, for example U.S. patent application Ser. No. 09/262,341 to Katzman et al., entitled “METHOD FOR THE DESIGN OF MULTIFOCAL OPTICAL ELEMENTS”, filed Mar. 4, 1999, the design is examined for its actual optical performance, and parameters of the design process can be adjusted in response to the optical performance, but this makes the design process indirect and time-consuming. Another typical approach in lens design is to use software means to trace a large number of rays through the lens and to deduce from the incident and refracted rays the classical aberrations of the lens. The surfaces of the lens are represented in terms of a fixed number of basic surfaces such as spheres, aspherics, torics and cylindrical surfaces. The ray tracing of a large number of rays requires a considerable computational effort, and the limited family of surfaces in which one optimizes the lens performance greatly restricts the optimization capabilities. Several methods were proposed to overcome the undesirable constraint of a limited family of optical surfaces. U.S. Pat. No. 4,613,217 to Fueter et al. proposes to represent an optical surface by splines. U.S. Pat. No. 5,886,766 to Kaga et al. similarly proposes a progressive ophthalmic lens consisting of three portions, at least one of which is divided into smaller pieces that are connected together by requiring that the surface is at least twice continuously differentiable along the interfaces joining two pieces. The spline representation, while offering an advantage over the limited class of surfaces mentioned above, has several drawbacks: the strict smoothness requirements along boundaries limits the space of parameters for the optimization process, the restriction to rectangular pieces, and thus to rectangular domains, limits the flexibility of the design, and it is not fully natural to prescribe boundary conditions on the lens surfaces and their slopes. U.S. patent application Ser. No. 09/262,341 to Katzman et al. describes a finite element method for the surface representation. The finite elements are patches of arbitrary polygonal shape, thus yielding flexible designs. The method of U.S. patent application Ser. No. 09/262,341 to Katzman et al. requires less differentiability along the lines joining the patches than that of U.S. Pat. No. 5,886,766 to Kaga et al. The common practice in the ophthalmic industry is to design and manufacture semi-finished lenses. The lens is manufactured with one surface that is given, for example a spherical or toric surface, and one surface that is designed. The given surface is later processed to meet the specific prescription of the client. Consequently most designs are limited in that they take into account only one surface of the lens, assuming, in general, a given spherical or toric other surface. Furthermore, the capability of improving the lens performance by designing both surfaces is not utilized. Recent technological developments enable better control of the manufacturing of both surfaces of ophthalmic lenses. Indeed, U.S. Pat. No. 5,771,089 to Barth, U.S. Pat. No. 5,784,144 to Kelch and U.S. patent application Ser. No. 09/262,341 to Katzman et al. propose designs in which both the front and back surfaces of the lens have flexible shapes. Further technological developments enable manufacturing of multi-surface ophthalmic lenses. U.S. patent application Ser. No. 09/262,341 to Katzman et al. proposes designs in which several of the surfaces have flexible shapes. In the design of multifocal ophthalmic lens, one usually selects the front (far from eye) surface to be progressive, while the back (close to eye) surface is either spherical or toric, where a toric surface might be needed to correct astigmatism. U.S. Pat. No. 2,878,721 to Kanolt and U.S. Pat. No. 6,019,470 to Mukaiyama et al. disclose ophthalmic lenses in which the back surface is a composition of a progressive surface and a toric surface. This composite design has a drawback that each of the two surfaces (toric and progressive) is designed separately, and thus the optical behavior of one of the surfaces may conflict with the optical behavior of the other. It is therefore beneficial to develop a method for designing a lens consisting of integral surfaces rather than composite surfaces. Such a method could then simultaneously consider all desired optical behavior. The actual performance of an ophthalmic lens depends not just on the lens itself, but on the full eye-plus-lens system. This becomes particularly important, for example, when the lens user suffers from astigmatism and/or presbyopia. Astigmatism is a condition in which the eye focuses differently in different directions. Presbyopia is a condition in which the eye loses some of its ability to accommodate, i.e. to focus sharply at nearby objects. The curvature of the lens of the eye changes as the eye focuses on objects at different distances from the eye. As people age, their eyes become less elastic and therefore can change the curvature of the lens only to a certain degree. The article by J. Loos, Ph. Slusallek and H.-P. Seidel, entitled “Using Wavefront Tracing for the Visualization and Optimization of Progressive Lenses”, An important characteristic of a lens is its prism. Prism measures the change in the direction of light rays as they are refracted by the lens. Essentially every lens has some level of prism. Sometimes a lens is processed in order to induce some desired prism for a variety of purposes. However, design methods taking prism into account are not known. There is provided in accordance with an embodiment of the present invention a method for designing at least one surface of an optical element. For each particular surface, a representation is created with a set of discrete points. Then for each discrete point in the set, nearest neighbor points from the set are selected, the particular surface is approximated in a vicinity of the discrete point by a polynomial of a predetermined order in two variables of the particular surface, and the coefficients of the polynomial are determined according to the selected nearest neighbor points without requiring continuity between polynomials for neighboring discrete points. There is also provided in accordance with an embodiment of the present invention a method for designing at least one surface of an optical element. The method includes the steps of representing the at least one surface using the method of unconstrained patches, choosing a function in parameters of the at least one surface, and optimizing the function with respect to the parameters. There is also provided in accordance with an embodiment of the present invention a method for designing at least one surface of an ophthalmic lens. The method includes the steps of representing the at least one surface with parameters, choosing a function in the parameters, and optimizing the function with respect to the parameters. Preferably, the ophthalmic lens is a multifocal progressive lens. Preferably, the function includes a term involving the difference between the astigmatism induced by the ophthalmic lens and a predetermined astigmatism distribution. The predetermined astigmatism distribution describes different astigmatism than that required by a prescription for the ophthalmic lens. Preferably, the function includes a term involving a power other than 2 of the astigmatism induced by the ophthalmic lens. Preferably, the function includes a term involving the difference between the prism induced by the ophthalmic lens and a predetermined prism distribution. Preferably, the function includes a term involving the difference between the gradient of a characteristic and the gradient of a predetermined characteristic distribution, where the characteristic is selected from a group including: power, astigmatism, and prism. Preferably, the function includes the square of the difference between the power induced by the ophthalmic lens and a predetermined power distribution and other terms related to the power induced by the ophthalmic lens. The other terms related to the power induced by the ophthalmic lens include terms of the form |P(l,m)−P Preferably, the function includes a term related to the thickness of the ophthalmic lens. There is also provided in accordance with an embodiment of the present invention a method for designing at least one surface of a multifocal optical element. The method includes the step of concurrently considering in a design of the at least one surface both toric and progressive portions of a prescription for the multifocal optical element. There is also provided in accordance with an embodiment of the present invention a method for designing a surface of an optical element. The method includes the steps of prescribing initial wavefronts, selecting an initial representation of the surface, the representation including parameters, and precomputing eikonal functions between points in the vicinity of the initial representation and points in the vicinity of the initial wavefronts. When optimizing a cost function dependent upon the parameters, a refracted wavefront for each of the initial wavefronts is calculated from the precomputed eikonal functions. Moreover, the step of optimizing includes the step of analytically computing derivatives of the cost function using the precomputed eikonal functions. Furthermore, if during the step of optimizing, a current representation of the surface varies too much from the initial representation, then eikonal functions between points in the vicinity of the current representation and points in the vicinity of the initial wavefronts are computed. There is provided in accordance with an embodiment of the present invention a method for designing a surface of an optical element. The method includes the steps of prescribing initial wavefronts, selecting an initial parameterized representation of the surface, choosing a function in the parameters, and optimizing the function with respect to the parameters. The step of optimizing includes the steps of calculating a refracted wavefront for each of the initial wavefronts and analytically computing derivatives of the function. There is provided in accordance with an embodiment of the present invention a method for designing at least one surface of an ophthalmic lens. The method includes the steps of representing the at least one surface by parameters, choosing an astigmatism distribution and an astigmatism direction distribution, choosing a function in the parameters and optimizing the function with respect to the parameters. The function includes a term involving the astigmatism distribution, the astigmatism direction distribution and the astigmatism induced by the ophthalmic lens. Preferably, the ophthalmic lens is a multifocal progressive lens. Preferably, the astigmatism distribution and the astigmatism direction distribution are determined from an eyeglass prescription. Preferably, the term is of the form where (l,m) index the parameters for the at least one surface, C The present invention will be understood and appreciated more fully from the following detailed description taken in conjunction with the appended drawings in which: FIG. 1 is a schematic illustration of an eye, and a lens having at least two surfaces, helpful in understanding the present invention; FIG. 2 is a flowchart illustration of a method for designing optical elements using wavefront calculations, according to an embodiment of the present invention; FIG. 3 is a schematic illustration of surfaces of a lens and of chief rays passing therethrough, according to an embodiment of the present invention; FIG. 4 is a schematic illustration of surfaces of a lens and of chief rays passing therethrough, according to an embodiment of the present invention, helpful in understanding the representation of localized wavefronts; FIG. 5 is a flowchart illustration of the optimization step of FIG. 2, according to an embodiment of the present invention; FIG. 6 is a schematic illustration of a chief ray and surfaces of an optical element, according to an embodiment of the present invention; FIG. 7 is a flowchart illustration of the eikonal method for an illustrative, non-limiting example, according to an embodiment of the present invention; FIG. 8 is a schematic illustration of the surface S Appendix A is a derivation of equations for point objects and wavefronts used in the eikonal method of FIG. The present invention is directed to a method for designing optical elements using wavefront calculations. An optical element is designed so that it converts a set of prescribed initial wavefronts into a set of associated wavefronts with desired properties. In general, it is not possible to solve the problem exactly, so an optimization method is used to produce the best optical element subject to the design constraints. The resulting designed surfaces of the optical element are integral surfaces and not composite surfaces. The present invention provides a method that enables design of a multi-surface optical element, in which any or all of the surfaces are designed, possibly simultaneously, using the method of the present invention. The present invention is directed to efficient optimization methods that provide the designer of the optical element with a large number of degrees of freedom to ensure a superior optical element. The degrees of freedom are the set of variables that define any of the surfaces of the optical element that need to be designed. It may be that some of the surfaces are fixed and cannot be changed. In that case, only the variables determining the non-fixed surfaces are considered to be degrees of freedom. In order to provide the designer with a large number of degrees of freedom, it is necessary to represent the surfaces efficiently and accurately. Applicants have found that it is advantageous to use several different surface representations. As will be described hereinbelow, some representations are useful for calculating wavefronts, while another representation is useful in an optimization process. In ideal situations, the surfaces are given explicitly in terms of well-known functions. Examples of such surfaces include spherical surfaces, aspherics, etc. In general, though, the designer should be allowed to design arbitrary surfaces, and therefore, in the present invention, discrete surfaces are used. Discrete surfaces are described in terms of finitely many points, for example, in the form of a set Q={p There are several efficient ways to describe discrete surfaces. Spline methods have the drawbacks mentioned in the background. U.S. patent application Ser. No. 09/262,341 to Katzman et al. describes a finite element method for describing discrete surfaces. Applicants have discovered that a surface representation, termed “unconstrained patches”, is highly efficient for the purpose of optical design. An arbitrary surface is described in terms of N discrete points, for example, in the form of the set Q mentioned hereinabove. In the vicinity of each discrete point p Therefore, each point p The present invention is particularly suitable for the design of ophthalmic spectacle lenses, including progressive addition lenses. Therefore, the detailed description that follows describes such an application, in which the initial wavefronts are refracted by the surfaces of the lens. However, it will be appreciated by persons skilled in the art that the present invention is equally applicable to the design of other optical elements including contact lenses and intraocular lenses and to the design of corneal shaping. Reference is now made to FIG. 1, which is a schematic illustration of an eye An initial surface is chosen for each of the unknown surfaces When ray If point object The eye has a natural aperture—the eye's pupil Since the shape of refracted wavefront Reference is also made to FIG. 2, which is a flowchart illustration of a method for designing optical elements using wavefront calculations, according to an embodiment of the present invention. Initial surfaces are chosen for the unknown surfaces of the optical element (step The designer of the lens then prescribes initial wavefronts (step The spherical wavefront A chief ray is constructed, as explained hereinabove with respect to FIG. 1, for each of the point objects (step The optimization process of the present invention requires repeated calculations of the intersection points of the chief rays with the surfaces of the optical element. When the surfaces are described discretely in a fixed global coordinate system, for example Cartesian coordinates or polar coordinates, as described hereinabove, this calculation is computationally intensive. Therefore, an alternative “ray representation”, in which the surfaces are represented by their distance from predetermined fixed points along the chief rays, is preferred. The difference between the previous representation and the ray representation is analogous to the difference between the Eulerian and Lagrangian formulations that are well known in the theory of mechanics. A ray representation of the known and initial surfaces Fixing the directions of the various segments of chief ray r(l,m) (using Snell's law), the intersection points q The set of values {t A ray representation can also be used in conjunction with other surface representations, for example in a finite elements method. Consider a system of rays parameterized by two discrete indices (l,m) and a single refractive surface S Once the surfaces have been represented by a ray representation, one needs to calculate the distortion of the initial localized wavefronts as they are refracted by the lens. A standard way to compute these distortions is to trace many rays through the lens. This method has the serious disadvantage that it requires the computation of the intersection of many rays with each of the lens surfaces, and the refraction of the rays at the intersection. Such calculations make this approach highly inefficient, since, in the optimization process, many computations of the distortion of localized wavefronts are made. Since, as mentioned hereinabove, only a small portion of each of the spherical wavefronts The localized wavefronts are represented (step According to an embodiment of the present invention, the calculation is based on a direct computation of the first few terms of the Taylor expansion of the localized wavefront as it is refracted by an optical surface. For this purpose, “front-associated” coordinate systems are defined. These are local coordinate systems having a {circumflex over (ζ)} axis pointing in the direction of the chief ray and variables ξ and η in the orthogonal plane. Since the wavefront is orthogonal to the chief ray, there are no linear terms in the Taylor expansion for the wavefront's local representation in these front-associated coordinates. Thus the leading order terms in the Taylor expansion for the wavefront are given by Equation 1:
where α According to an embodiment of the present invention, only quadratic terms in the local Taylor expansion are considered. This effectively acts to restrict the wavefront to the portion that is transmitted through the pupil. These quadratic terms are related to the principal curvatures of the wavefront and their directions. The relations between the Taylor coefficients and the power and astigmatism of the wavefront are given by Equation 2:
The variables ξ and η are defined above with respect to an arbitrary orthogonal coordinate system perpendicular to the chief ray. In some applications, as will be described in more detail hereinbelow, it is preferred to choose a specific coordinate system in the plane perpendicular to the chief ray. For example, Applicants have discovered that it might be preferable to rotate the ({circumflex over (ξ)},{circumflex over (η)})coordinate system so that the new axes will be along the directions of the principal curvatures of the wavefront. It will be appreciated that in this rotated coordinate system, the Taylor coefficient α Therefore, the triplets {[α for the localized incoming wavefront, and the front-associated coordinate system ({circumflex over (ξ)} for the localized refracted wavefront. It is these Taylor coefficients for the localized refracted wavefronts for each chief ray r(l,m) that need to be calculated. Since the localized wavefronts are of finite extent, it is expected that they will overlap somewhat. A given area on the lens may focus wavefronts from one point object differently than it focuses wavefronts from another point object. Therefore it may happen that some lens shapes are advantageous for viewing some objects but not for viewing other objects. The optimization process compromises between the requirements of good lens performance for all the point objects. In order to compute a lens providing high quality vision, the designer determines a cost function to be optimized (step In an embodiment of the present invention, the cost function G is written in the form given in Equation 3: where (l,m) are the parameters used to describe the lens surfaces, D is the domain in which l and m are defined, w
where T The designer selects the factors U Required Power. For this purpose, the designer predetermines a function P Required Astigmatism. Every progressive lens has some level of astigmatism C(l,m). In many applications, the designer attempts to minimize this astigmatism. For example, this is a design goal when designing semi-finished lenses or when designing a customized lens according to the specific prescription of a client where the prescription has no level of astigmatism. Such situations are termed “astigmatic-free” designs. In other applications, the designer wants to achieve a lens with a predetermined astigmatism distribution. For example, this is a design goal when designing a customized lens according to the specific prescription of a client where the prescription has some level of astigmatism. Such situations are termed “astigmatic” designs. It is appreciated that the astigmatism in astigmatic designs includes information on the required directions of astigmatism. Astigmatic-free designs can be accomplished, for example, by using a term of the form C(l,m) In another preferred embodiment, a weighted combination of terms of the form |C(l,m)−C Terms U In an embodiment, the astigmatism direction distribution is used to fix the ({circumflex over (ξ)},{circumflex over (η)}) coordinate system in the orthogonal plane for each chief ray. Using this choice, the Taylor coefficient α In another preferred embodiment, combinations of terms of the form for various choices of β are used in the cost function, where w Cosmetic considerations. One of the criteria in the ophthalmic industry for the quality of a lens is its overall appearance, and in particular its thickness. Since the optimization methods of the present invention take the full structure of the lens into account, together with its constituent surfaces, Applicants have discovered that it can be useful to add to the cost function a term that is proportional to the lens thickness at the point (l,m). Required prism. The local behavior of each wavefront determines the local prism of the lens. In fact, the prism is already determined by the deflection of the chief ray. Applicants have realized that it is useful to predetermine a function Prism Variations in the lens characteristics. Applicants have realized that the lens performance can be improved by preventing too rapid variations in the lens characteristics. Alternatively, the designer may wish to obtain a prescribed distribution of the variations. For every lens characteristic, denoted by CH, the designer adds a weighted combination of terms of the form |∇CH(l,m)−∇CH The design method based on the cost function G can be used to design surfaces that are progressive, astigmatic and prismatic. Moreover, any combination of the progression of the optical power, the astigmatism of the lens, and the prism of the lens are designed simultaneously and concurrently, thus leading to integral surfaces. It will be appreciated that the design of integral surfaces is superior to the prior art, as described in U.S. Pat. No. 2,878,721 to Kanolt and U.S. Pat. No. 6,019,470 to Mukaiyama et al., where two separate surfaces are designed, one for the optical power progression, and one for the required astigmatism, and then composed to obtain a composite surface. Consequently, in a lens with a composite surface, one or more of the constituents of the composite surface may damage some of the optimal properties of the other constituents. While in the design of a composite surface there is no correlation between the two parts (progressive and toric), the design of an integral surface takes both of them into account simultaneously, thus ensuring an overall optimization. In another example, the thickness of a toric contact lens is increased in certain regions of the lens so that when worn, it will stay on the eye in a particular orientation. The prism of the contact lens needs to be adjusted in those regions in order to compensate for the added thickness. Thus, it is beneficial to design one or more integral surfaces using a design method that takes the astigmatism and the prism into account simultaneously and concurrently. In a further example, there are design requirements for lenses having progressive power, astigmatism and prism. The design method described herein enables the design of an integral surface while concurrently taking into account all of these requirements. Once the cost function has been chosen, it needs to be optimized (step Many optimization methods are known in the art. For example, common techniques include the conjugate gradient method and the Newton method. Essentially all methods are iterative. Reference is now made to FIG. 5, which is a flowchart illustration of the optimization step When the Rth step in the iteration begins, each unknown surface S In an embodiment, the Taylor coefficients for the localized refracted wavefronts beyond the optical element are calculated using a localized front refraction method (step is constructed at the intersection point of the chief ray with the surface S By differentiating Snell's law, the Taylor coefficients of the localized wavefront after refraction by the surface S A formulation of the localized front refraction method is given in J. Kneisly, “Local curvature of wavefronts in an optical system”, JOSA 54, 1964, pp. 229-235. A formula is developed that directly connects the principal curvatures and directions of a refracted wavefront with the principal curvatures and directions of the wavefront before refraction and the principal curvatures and directions of the surface at the point of refraction. Applicants have discovered a novel method, based on eikonal functions, for calculating the localized refracted wavefronts beyond the optical element. Eikonal functions are well known classical objects in optics. O.N. Stavroudis, in terms of the free parameters. The eikonal method (step The calculation of the shape of the localized wavefront after refraction by all surfaces of the optical element, using the eikonal method or the localized front refraction method or some other suitable method, results in localized refracted wavefronts beyond the optical element (step Once the localized refracted wavefronts have been calculated, the cost function can be evaluated (step Each iteration step requires the calculation of the derivative of the cost function with respect to all its variables. In fact, popular optimization methods, such as the Newton method, require also the computation of the second derivatives. Since a flexible design includes a very large number of free variables, and since the computation of derivatives is required for each iteration, it is desirable to look for efficient methods of computing these derivatives. One possibility is to compute the derivatives numerically. For this purpose, one slightly varies a certain parameter and evaluates the resulting change in the cost function. Since the lens surfaces and the refraction of the wavefronts by them are characterized by local properties, there is no need to recompute the transfer of all of the wavefronts for each such small local variation. This localization accelerates the computation of the derivatives, and hence the entire optimization process. Another possibility is to compute the derivatives analytically, which can accelerate the optimization process dramatically. Applicants discovered that the eikonal method (step The design method begins with an initial choice for S For each chief ray r(l,m) a local orthogonal coordinate system having one of its axes in the direction of {circumflex over (τ)}(l,m) is defined (step The eikonal function E is classically defined as the optical distance between two points. For the spherical wavefront emitted by the point object o In the optimization process, the surface S It is necessary at each step of the optimization process to calculate the shape of the refracted wavefront at the surface S where the Taylor coefficients are denoted by denotes t However, the Taylor coefficients need to be determined in advance. Fermat's principle is used to calculate the value of the eikonal function between any point o in the vicinity of o The design process consists, therefore, of finding for each (l,m) the values t The optimization process is iterative. Consider the R th step in the iteration. The surface S Step
where the 2-dimensional vectors ε for n=1,2. The vectors ε and also involve the quantities which are defined as follows: and Equation 5 is an equation for the new point object o The coefficients of the local wavefront are given explicitly by:
where α of the refracted wavefront. Equation 11 involves the following 3×3 matrix W where t where the canonical quantities are defined as follows: Equation 5 and Equation 11, which are derived in Appendix A, are the fundamental equations in the novel eikonal method. They provide explicit expressions for the point objects o that can be precomputed at the initial stage and then stored in memory. It might happen that during the iterative optimization process the surface S It will be appreciated by persons skilled in the art that the present invention is not limited by what has been particularly shown and described herein above. Rather the scope of the invention is defined by the claims that follow Appendix A: This appendix is a derivation of Equations 5 and 11, relating the point objects o Reference is now made to FIG. 8, which is a schematic illustration of the surface S In order that the wavefront originating at the point object o The eikonal function E in the plane PL is now expressed in the orthogonal local coordinate system ({circumflex over (ξ)} A simple geometrical calculation then provides that the orthogonal projection of the points t where the geometrical quantities are defined in Equation 8: The condition on the point object o The Euclidean distance between a point t It follows that, to linear order, the optical distance between point t Since the condition of Equation A2 holds, to linear order, the following equation is obtained: In particular, the equivalence holds for the projected points defined in Equation A1: The left side of Equation A4 is precisely the difference between the eikonal values at t where is defined in Equation 9, and is defined in Equation 10: Equation A5 is a system of 9 equations for the unknown point object o Equation A6 is applied at the projected points, and using the linear-order optical distance, the following is obtained: where p,q=0,±1. Eq. A7 forms an overdetermined system of 9 equations in the two unknowns considering for the moment the unknown point object o The normal equations for the least squares method can be written explicitly. It is convenient at this stage to introduce for each quadruple l,m,p,q the canonical 2×2 matrices: The matrix B The normal equations for the least squares approximation of Equation A7 are:
where and where the 2-dimensional vectors ε and for n=1,2. Imposing now the condition ∇σ(
It is useful at this stage to point out certain computational aspects. Firstly, the geometrical matrices and B depend only on the directions of the chief rays and are therefore computed only once (at the initial stage) and stored in memory. Finally, the vectors ε Determining the point object o
The local quadratic expansion of Equation A11 is precisely the local expansion for the wavefront passing through t By substituting Equation A1 into Equation A11 and using the fact that one obtains the following: By defining the following canonical quantities as in Equation 13 in the detailed description: Equation A12 can now be written compactly in the following form: for p,q=0,±1. Eq. A13 forms an overdetermined system of 9 equations in 3 unknowns (α where t The normal equations are then given by:
where α of the refracted wavefront. The coefficients of the local wavefront are given explicitly by Equation 11 of the detailed description:
Patent Citations
Non-Patent Citations
Referenced by
Classifications
Legal Events
Rotate |